Furthermore, mixed-monotonicity has also proven to be very beneficial for system analysis and control.
For instance, if mixed-monotonicity holds, it can be concluded that the original system has global asymptotic stability by proving the nonexistence of equilibria of the  lifted system, except in a certain lower-dimensional subspace [1]}, [2]}.  Moreover, forward invariant and attractive sets of the original system
can be identified [3]} and reachable sets of the original system can be efficiently  over-approximated  and used for state estimation and abstraction-based control synthesis [3]}, [5]}, [6]}, [7]}.
 However, the usefulness of these system lifting techniques for analysis and control is highly dependent on the tightness of the decomposition approaches, their computational tractability [8]}, [9]} and their applicability to a broad class of systems; therefore, the capability to compute or construct tractable and tight mixed-monotone decomposition functions for a broad range of nonlinear, uncertain and constrained systems is of great interest and will have a significant impact.
